Non - Ergodic Dynamics of the 2 D Random - Phase Sine - Gordon Model : Applications to Vortex - Glass Arrays and Disordered - Substrate Surfaces

The dynamics of the random-phase sine-Gordon model, which describes 2D vortex-glass arrays and crystalline surfaces on disordered substrates, is investigated using the self-consistent Hartree approximation. The fluctuation-dissipation theorem is violated below the critical temperature T c for large time t > t * where t * diverges in the thermodynamic limit. While above T c the averaged autocorrelation function diverges as T ln(t), for T < T c it approaches a finite value q * ∼ 1/(T c − T) as q(t) = q * − c (t/t *) −ν (for t → t *) where ν is a temperature-dependent exponent. On larger time scales t > t * the dynamics becomes non-ergodic. The static correlations behave as ∼ T ln | x| for T > T c and for T < T c when x < ξ * with ξ * ∼ exp {A/(T c − T)}. For scales